Subgroups — C₃₁×#61916 #21518675

Order 7936.

1696 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3401	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3398	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3394	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3373	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3369	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3359	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3225	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3214	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3125	C₂	1	4	yes	maximal, complemented
13	C₃₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2985	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#563	—	2	2	no	—
17	C₃₁×#559	C₂²	1	2	yes	—
18	C₃₁×#559	C₂²	1	2	yes	—
19	Q₈²×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#557	C₂²	1	2	yes	—
21	C₃₁×#556	—	2	1	no	—
22	C₃₁×#556	—	2	1	no	—
23	C₃₁×#556	C₂²	1	4	yes	—
24	C₃₁×#556	C₂²	1	2	yes	—
25	C₃₁×#554	—	2	1	no	—
26	C₃₁×#552	—	2	2	no	—
27	D₄×Q₈×C₃₁	—	2	1	no	—
28	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#542	—	2	2	no	—
30	C₃₁×#542	C₂²	1	4	yes	—
31	C₃₁×#539	—	2	2	no	—
32	C₃₁×#536	—	2	2	no	—
33	C₃₁×#536	C₂²	1	2	yes	—
34	C₃₁×#533	C₂²	1	1	yes	characteristic
35	C₃₁×#533	C₂²	1	1	yes	characteristic
36	C₃₁×#532	C₂²	1	1	yes	characteristic
37	C₃₁×#532	C₂²	1	1	yes	characteristic
38	C₃₁×#529	—	2	2	no	—
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#497	C₂²	1	4	yes	—
42	C₃₁×#496	C₂²	1	2	yes	—
43	C₃₁×#495	C₂²	1	4	yes	—
44	C₃₁×#494	C₂²	1	1	yes	characteristic
45	C₃₁×#492	C₂²	1	1	yes	characteristic
46	C₃₁×#489	C₂²	1	2	yes	—
47	C₃₁×#489	C₂²	1	4	yes	—
48	C₃₁×#487	C₂²	1	1	yes	characteristic
49	C₃₁×#487	C₂²	1	4	yes	—
50	C₃₁×#486	C₂²	1	1	yes	characteristic
51	C₃₁×#483	C₂²	1	2	yes	—
52	C₃₁×#481	C₂²	1	2	yes	—
53	C₃₁×#476	C₂²	1	4	yes	—
54	C₃₁×#475	C₂²	1	2	yes	—
55	C₃₁×#475	C₂²	1	2	yes	—
56	C₃₁×#475	C₂²	1	4	yes	—
57	C₃₁×#464	C₂²	1	4	yes	—
58	C₃₁×#464	C₂²	1	4	yes	—
59	C₃₁×#464	C₂²	1	4	yes	—
60	C₃₁×#463	C₂²	1	4	yes	—
61	C₃₁×#463	C₂²	1	4	yes	—
62	C₃₁×#462	C₂²	1	2	yes	—
63	C₃₁×#462	C₂²	1	4	yes	—
64	C₃₁×#461	C₂²	1	2	yes	—
65	C₃₁×#461	C₂²	1	4	yes	—
66	C₃₁×#460	C₂²	1	2	yes	—
67	C₃₁×#459	C₂²	1	2	yes	—
68	C₃₁×#452	C₂²	1	4	yes	—
69	C₃₁×#451	C₂²	1	2	yes	—
70	C₃₁×#449	C₂²	1	4	yes	—
71	C₃₁×#448	C₂²	1	2	yes	—
72	C₃₁×#446	C₂²	1	4	yes	—
73	C₃₁×#446	C₂²	1	2	yes	—
74	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
75	C₃₁×#442	C₂²	1	4	yes	—
76	C₃₁×#441	C₂²	1	4	yes	—
77	C₃₁×#441	C₂²	1	4	yes	—
78	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
79	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
80	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
81	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
82	C₃₁×#433	C₂²	1	4	yes	—
83	C₃₁×#432	C₂²	1	1	yes	characteristic
84	C₃₁×#431	C₂²	1	1	yes	characteristic
85	C₂×#41×C₃₁	—	2	1	no	—
86	C₂×#41×C₃₁	C₂³	1	2	yes	—
87	C₂²×Q₈×C₃₁	—	2	2	no	—
88	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
91	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
92	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
93	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
95	C₂×D₈×C₃₁	C₂³	1	4	yes	—
96	C₃₁×C₄⋊Q₈	—	2	4	no	—
97	C₃₁×C₄⋊Q₈	—	2	2	no	—
98	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
99	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
101	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
102	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
103	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
104	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
105	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
106	C₃₁×#127	C₂³	1	1	yes	characteristic
107	C₃₁×#127	C₂³	1	1	yes	characteristic
108	C₃₁×#126	—	2	4	no	—
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#126	—	2	4	no	—
111	C₃₁×#126	—	2	4	no	—
112	C₃₁×#125	—	2	2	no	—
113	C₃₁×#125	—	2	4	no	—
114	C₃₁×#125	—	2	1	no	—
115	C₃₁×#125	D₄	1	2	yes	—
116	C₃₁×#124	—	2	2	no	—
117	C₃₁×#124	—	2	2	no	—
118	C₃₁×#124	—	2	4	no	—
119	C₃₁×#124	—	2	1	no	—
120	C₃₁×#124	C₂³	1	4	yes	—
121	C₃₁×#124	C₂³	1	4	yes	—
122	C₃₁×#124	C₂³	1	2	yes	—
123	C₃₁×#123	—	2	4	no	—
124	C₃₁×#123	—	2	4	no	—
125	C₃₁×#123	—	2	4	no	—
126	C₃₁×#123	—	2	4	no	—
127	C₃₁×C₂²⋊Q₈	—	2	2	no	—
128	C₃₁×C₂²⋊Q₈	—	2	4	no	—
129	C₃₁×C₂²⋊Q₈	—	2	1	no	—
130	C₃₁×C₂²⋊Q₈	—	2	2	no	—
131	C₃₁×C₂²⋊Q₈	—	2	2	no	—
132	C₃₁×C₂²⋊Q₈	—	2	2	no	—
133	C₃₁×C₂²⋊Q₈	—	2	4	no	—
134	C₃₁×C₂²⋊Q₈	—	2	4	no	—
135	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
136	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
137	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
138	C₃₁×#121	—	2	1	no	—
139	C₃₁×#121	—	2	1	no	—
140	C₃₁×#121	—	2	1	no	—
141	C₃₁×#121	C₂³	1	2	yes	—
142	C₄×Q₈×C₃₁	—	2	2	no	—
143	C₄×Q₈×C₃₁	—	2	2	no	—
144	C₄×Q₈×C₃₁	—	2	1	no	—
145	C₄×Q₈×C₃₁	—	2	1	no	—
146	C₄×Q₈×C₃₁	—	2	1	no	—
147	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
148	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
149	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
153	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
154	C₄×D₄×C₃₁	—	2	2	no	—
155	C₄×D₄×C₃₁	—	2	2	no	—
156	C₄×D₄×C₃₁	—	2	4	no	—
157	C₄×D₄×C₃₁	—	2	1	no	—
158	C₄×D₄×C₃₁	—	2	1	no	—
159	C₄×D₄×C₃₁	—	2	1	no	—
160	C₄×D₄×C₃₁	C₂³	1	2	yes	—
161	C₄×D₄×C₃₁	C₂³	1	4	yes	—
162	C₄×D₄×C₃₁	C₂³	1	4	yes	—
163	C₃₁×#117	—	2	1	no	—
164	C₃₁×#117	—	2	2	no	—
165	C₃₁×#117	—	2	4	no	—
166	C₃₁×#117	—	2	2	no	—
167	C₃₁×#117	C₂³	1	2	yes	—
168	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
171	C₂×C₄²×C₃₁	—	2	1	no	—
172	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
173	C₃₁×#106	C₂³	1	2	yes	—
174	C₃₁×#106	C₂³	1	2	yes	—
175	C₃₁×#106	C₂³	1	4	yes	—
176	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
177	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
178	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
179	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
180	C₃₁×#103	C₂³	1	4	yes	—
181	C₃₁×#103	C₂³	1	2	yes	—
182	C₃₁×#103	C₂³	1	2	yes	—
183	C₃₁×#103	C₂³	1	4	yes	—
184	C₃₁×#103	C₂³	1	4	yes	—
185	C₃₁×#103	C₂³	1	4	yes	—
186	C₃₁×#102	C₂³	1	2	yes	—
187	C₃₁×#102	C₂³	1	2	yes	—
188	C₃₁×#102	C₂³	1	4	yes	—
189	C₃₁×#102	C₂³	1	4	yes	—
190	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
191	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
192	C₃₁×#97	C₂³	1	2	yes	—
193	C₃₁×#97	C₂³	1	4	yes	—
194	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
195	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
196	C₄×C₈×C₃₁	C₂³	1	4	yes	—
197	#41×C₃₁	—	4	4	no	—
198	#41×C₃₁	—	8	2	no	—
199	C₂×Q₈×C₃₁	—	2	1	no	—
200	C₂×Q₈×C₃₁	—	2	1	no	—
201	C₂×Q₈×C₃₁	—	2	4	no	—
202	C₂×Q₈×C₃₁	—	2	1	no	—
203	C₂×Q₈×C₃₁	—	4	2	no	—
204	C₂×Q₈×C₃₁	—	8	4	no	—
205	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
206	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
207	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
208	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
209	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
210	C₂×D₄×C₃₁	—	2	2	no	—
211	C₂×D₄×C₃₁	—	2	2	no	—
212	C₂×D₄×C₃₁	—	2	1	no	—
213	C₂×D₄×C₃₁	—	2	1	no	—
214	C₂×D₄×C₃₁	—	2	1	no	—
215	C₂×D₄×C₃₁	—	2	1	no	—
216	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
217	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₃₁	—	2	1	no	—
219	C₂²×C₄×C₃₁	—	2	1	no	—
220	C₂²×C₄×C₃₁	—	2	2	no	—
221	C₂²×C₄×C₃₁	—	2	2	no	—
222	C₂²×C₄×C₃₁	—	2	4	no	—
223	C₂²×C₄×C₃₁	—	2	4	no	—
224	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
227	Q₁₆×C₃₁	—	4	4	no	—
228	Q₁₆×C₃₁	—	4	4	no	—
229	SD₁₆×C₃₁	—	4	4	no	—
230	SD₁₆×C₃₁	—	4	4	no	—
231	SD₁₆×C₃₁	—	4	2	no	—
232	SD₁₆×C₃₁	—	4	2	no	—
233	D₈×C₃₁	—	4	4	no	—
234	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
235	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
236	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
237	C₄⋊C₄×C₃₁	—	2	1	no	—
238	C₄⋊C₄×C₃₁	—	2	4	no	—
239	C₄⋊C₄×C₃₁	—	2	2	no	—
240	C₄⋊C₄×C₃₁	—	2	2	no	—
241	C₄⋊C₄×C₃₁	—	2	1	no	—
242	C₄⋊C₄×C₃₁	—	2	1	no	—
243	C₄⋊C₄×C₃₁	—	2	4	no	—
244	C₄⋊C₄×C₃₁	—	2	1	no	—
245	C₄⋊C₄×C₃₁	—	2	1	no	—
246	C₄⋊C₄×C₃₁	—	2	1	no	—
247	C₄⋊C₄×C₃₁	—	2	1	no	—
248	C₄⋊C₄×C₃₁	—	2	2	no	—
249	C₄⋊C₄×C₃₁	—	2	4	no	—
250	C₄⋊C₄×C₃₁	—	2	4	no	—
251	C₄⋊C₄×C₃₁	—	2	4	no	—
252	C₄⋊C₄×C₃₁	—	2	2	no	—
253	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
254	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
255	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
256	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
257	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
259	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
260	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
261	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
262	C₂²⋊C₄×C₃₁	—	2	2	no	—
263	C₂²⋊C₄×C₃₁	—	2	2	no	—
264	C₂²⋊C₄×C₃₁	—	2	4	no	—
265	C₂²⋊C₄×C₃₁	—	2	4	no	—
266	C₂²⋊C₄×C₃₁	—	2	4	no	—
267	C₂²⋊C₄×C₃₁	—	2	2	no	—
268	C₂²⋊C₄×C₃₁	—	2	2	no	—
269	C₂²⋊C₄×C₃₁	—	2	4	no	—
270	C₂²⋊C₄×C₃₁	—	2	4	no	—
271	C₂²⋊C₄×C₃₁	—	2	2	no	—
272	C₂²⋊C₄×C₃₁	—	2	4	no	—
273	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
274	C₄²×C₃₁	—	2	2	no	—
275	C₄²×C₃₁	—	2	2	no	—
276	C₄²×C₃₁	—	2	4	no	—
277	C₄²×C₃₁	—	2	2	no	—
278	C₄²×C₃₁	—	2	4	no	—
279	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
281	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
282	C₄²×C₃₁	C₂⁴	1	4	yes	—
283	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
284	#61916	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
285	C₂³×C₃₁	—	2	2	no	—
286	C₂³×C₃₁	—	2	2	no	—
287	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
288	Q₈×C₃₁	—	2	4	no	—
289	Q₈×C₃₁	—	2	2	no	—
290	Q₈×C₃₁	—	2	4	no	—
291	Q₈×C₃₁	—	2	4	no	—
292	Q₈×C₃₁	—	2	2	no	—
293	Q₈×C₃₁	—	4	2	no	—
294	Q₈×C₃₁	—	4	2	no	—
295	Q₈×C₃₁	—	4	2	no	—
296	Q₈×C₃₁	—	8	4	no	—
297	Q₈×C₃₁	—	8	1	no	—
298	D₄×C₃₁	—	2	2	no	—
299	D₄×C₃₁	—	2	4	no	—
300	D₄×C₃₁	—	2	2	no	—
301	D₄×C₃₁	—	4	2	no	—
302	D₄×C₃₁	—	4	4	no	—
303	D₄×C₃₁	—	4	2	no	—
304	D₄×C₃₁	—	8	1	no	—
305	D₄×C₃₁	—	8	1	no	—
306	D₄×C₃₁	—	8	2	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	2	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	2	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	2	no	—
313	C₂×C₄×C₃₁	—	2	2	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	4	1	no	—
316	C₂×C₄×C₃₁	—	4	4	no	—
317	C₂×C₄×C₃₁	—	4	2	no	—
318	C₂×C₄×C₃₁	—	4	4	no	—
319	C₂×C₄×C₃₁	—	8	4	no	—
320	C₂×C₄×C₃₁	—	8	4	no	—
321	C₂×C₄×C₃₁	—	8	1	no	—
322	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
325	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
326	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
328	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
330	C₈×C₃₁	—	2	4	no	—
331	C₈×C₃₁	—	2	2	no	—
332	C₈×C₃₁	—	2	2	no	—
333	#3550	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#3401	C₂×C₃₁	1	4	yes	complemented
336	#3398	C₂×C₃₁	1	2	yes	complemented
337	#3394	C₂×C₃₁	1	2	yes	complemented
338	#3373	C₂×C₃₁	1	2	yes	complemented
339	#3369	C₂×C₃₁	1	4	yes	complemented
340	#3359	C₂×C₃₁	1	2	yes	complemented
341	#3225	C₂×C₃₁	1	4	yes	complemented
342	#3214	C₂×C₃₁	1	1	yes	characteristic
343	#3210	C₂×C₃₁	1	1	yes	characteristic, complemented
344	#3125	C₂×C₃₁	1	4	yes	complemented
345	#3113	C₂×C₃₁	1	1	yes	characteristic, complemented
346	#3103	C₂×C₃₁	1	1	yes	characteristic, complemented
347	#2985	C₂×C₃₁	1	1	yes	characteristic, complemented
348	C₂²×C₃₁	—	2	1	no	—
349	C₂²×C₃₁	—	2	1	no	—
350	C₂²×C₃₁	—	2	1	no	—
351	C₂²×C₃₁	—	4	2	no	—
352	C₂²×C₃₁	—	4	2	no	—
353	C₂²×C₃₁	—	4	2	no	—
354	C₂²×C₃₁	—	4	2	no	—
355	C₂²×C₃₁	—	4	2	no	—
356	C₂²×C₃₁	—	4	2	no	—
357	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
358	C₄×C₃₁	—	2	1	no	—
359	C₄×C₃₁	—	2	1	no	—
360	C₄×C₃₁	—	2	4	no	—
361	C₄×C₃₁	—	2	1	no	—
362	C₄×C₃₁	—	2	1	no	—
363	C₄×C₃₁	—	2	4	no	—
364	C₄×C₃₁	—	4	2	no	—
365	C₄×C₃₁	—	4	2	no	—
366	C₄×C₃₁	—	4	2	no	—
367	C₄×C₃₁	—	4	2	no	—
368	C₄×C₃₁	—	4	2	no	—
369	C₄×C₃₁	—	4	2	no	—
370	C₄×C₃₁	#577	1	2	yes	—
371	C₄×C₃₁	#585	1	2	yes	—
372	#563	—	2	2	no	—
373	#559	C₂²×C₃₁	1	2	yes	—
374	#559	C₂²×C₃₁	1	2	yes	—
375	Q₈²	C₂²×C₃₁	1	2	yes	—
376	#557	C₂²×C₃₁	1	2	yes	—
377	#556	—	2	1	no	—
378	#556	—	2	1	no	—
379	#556	C₂²×C₃₁	1	4	yes	—
380	#556	C₂²×C₃₁	1	2	yes	—
381	#554	—	2	1	no	—
382	#552	—	2	2	no	—
383	D₄×Q₈	—	2	1	no	—
384	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
385	#542	—	2	2	no	—
386	#542	C₂²×C₃₁	1	4	yes	—
387	#539	—	2	2	no	—
388	#536	—	2	2	no	—
389	#536	C₂²×C₃₁	1	2	yes	—
390	#533	C₂²×C₃₁	1	1	yes	characteristic
391	#533	C₂²×C₃₁	1	1	yes	characteristic
392	#532	C₂²×C₃₁	1	1	yes	characteristic
393	#532	C₂²×C₃₁	1	1	yes	characteristic
394	#529	—	2	2	no	—
395	#520	C₂²×C₃₁	1	2	yes	—
396	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
397	#497	C₂²×C₃₁	1	4	yes	—
398	#496	C₂²×C₃₁	1	2	yes	—
399	#495	C₂²×C₃₁	1	4	yes	—
400	#494	C₂²×C₃₁	1	1	yes	characteristic
401	#492	C₂²×C₃₁	1	1	yes	characteristic
402	#489	C₂²×C₃₁	1	2	yes	—
403	#489	C₂²×C₃₁	1	4	yes	—
404	#487	C₂²×C₃₁	1	1	yes	characteristic
405	#487	C₂²×C₃₁	1	4	yes	—
406	#486	C₂²×C₃₁	1	1	yes	characteristic
407	#483	C₂²×C₃₁	1	2	yes	—
408	#481	C₂²×C₃₁	1	2	yes	—
409	#476	C₂²×C₃₁	1	4	yes	—
410	#475	C₂²×C₃₁	1	2	yes	—
411	#475	C₂²×C₃₁	1	2	yes	—
412	#475	C₂²×C₃₁	1	4	yes	—
413	#464	C₂²×C₃₁	1	4	yes	—
414	#464	C₂²×C₃₁	1	4	yes	—
415	#464	C₂²×C₃₁	1	4	yes	—
416	#463	C₂²×C₃₁	1	4	yes	—
417	#463	C₂²×C₃₁	1	4	yes	—
418	#462	C₂²×C₃₁	1	2	yes	—
419	#462	C₂²×C₃₁	1	4	yes	—
420	#461	C₂²×C₃₁	1	2	yes	—
421	#461	C₂²×C₃₁	1	4	yes	—
422	#460	C₂²×C₃₁	1	2	yes	—
423	#459	C₂²×C₃₁	1	2	yes	—
424	#452	C₂²×C₃₁	1	4	yes	—
425	#451	C₂²×C₃₁	1	2	yes	—
426	#449	C₂²×C₃₁	1	4	yes	—
427	#448	C₂²×C₃₁	1	2	yes	—
428	#446	C₂²×C₃₁	1	4	yes	—
429	#446	C₂²×C₃₁	1	2	yes	—
430	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
431	#442	C₂²×C₃₁	1	4	yes	—
432	#441	C₂²×C₃₁	1	4	yes	—
433	#441	C₂²×C₃₁	1	4	yes	—
434	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
435	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
436	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
437	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
438	#433	C₂²×C₃₁	1	4	yes	—
439	#432	C₂²×C₃₁	1	1	yes	characteristic
440	#431	C₂²×C₃₁	1	1	yes	characteristic
441	C₂×C₃₁	—	4	1	no	—
442	C₂×C₃₁	—	8	2	no	—
443	C₂×C₃₁	—	8	2	no	—
444	C₂×C₃₁	#3487	1	1	yes	characteristic
445	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
446	C₂×C₃₁	#3586	1	1	yes	characteristic
447	C₂×#41	—	2	1	no	—
448	C₂×#41	C₂³×C₃₁	1	2	yes	—
449	C₂²×Q₈	—	2	2	no	—
450	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
452	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
453	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
454	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
455	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
456	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
457	C₂×D₈	C₂³×C₃₁	1	4	yes	—
458	C₄⋊Q₈	—	2	4	no	—
459	C₄⋊Q₈	—	2	2	no	—
460	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
461	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
462	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
463	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
464	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
465	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
466	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
467	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
468	#127	C₂³×C₃₁	1	1	yes	characteristic
469	#127	C₂³×C₃₁	1	1	yes	characteristic
470	#126	—	2	4	no	—
471	#126	—	2	4	no	—
472	#126	—	2	4	no	—
473	#126	—	2	4	no	—
474	#125	—	2	2	no	—
475	#125	—	2	4	no	—
476	#125	—	2	1	no	—
477	#125	D₄×C₃₁	1	2	yes	—
478	#124	—	2	2	no	—
479	#124	—	2	2	no	—
480	#124	—	2	4	no	—
481	#124	—	2	1	no	—
482	#124	C₂³×C₃₁	1	2	yes	—
483	#124	C₂³×C₃₁	1	4	yes	—
484	#124	C₂³×C₃₁	1	4	yes	—
485	#123	—	2	4	no	—
486	#123	—	2	4	no	—
487	#123	—	2	4	no	—
488	#123	—	2	4	no	—
489	C₂²⋊Q₈	—	2	2	no	—
490	C₂²⋊Q₈	—	2	4	no	—
491	C₂²⋊Q₈	—	2	1	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	2	no	—
494	C₂²⋊Q₈	—	2	2	no	—
495	C₂²⋊Q₈	—	2	4	no	—
496	C₂²⋊Q₈	—	2	4	no	—
497	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
498	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
499	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
500	#121	—	2	1	no	—
501	#121	—	2	1	no	—
502	#121	—	2	1	no	—
503	#121	C₂³×C₃₁	1	2	yes	—
504	C₄×Q₈	—	2	2	no	—
505	C₄×Q₈	—	2	2	no	—
506	C₄×Q₈	—	2	1	no	—
507	C₄×Q₈	—	2	1	no	—
508	C₄×Q₈	—	2	1	no	—
509	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
510	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
511	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
512	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
513	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
514	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
515	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
516	C₄×D₄	—	2	2	no	—
517	C₄×D₄	—	2	2	no	—
518	C₄×D₄	—	2	4	no	—
519	C₄×D₄	—	2	1	no	—
520	C₄×D₄	—	2	1	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	C₂³×C₃₁	1	4	yes	—
523	C₄×D₄	C₂³×C₃₁	1	2	yes	—
524	C₄×D₄	C₂³×C₃₁	1	4	yes	—
525	#117	—	2	1	no	—
526	#117	—	2	2	no	—
527	#117	—	2	4	no	—
528	#117	—	2	2	no	—
529	#117	C₂³×C₃₁	1	2	yes	—
530	C₂×C₄⋊C₄	—	2	1	no	—
531	C₂×C₄⋊C₄	—	2	2	no	—
532	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
533	C₂×C₄²	—	2	1	no	—
534	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
535	#106	C₂³×C₃₁	1	2	yes	—
536	#106	C₂³×C₃₁	1	2	yes	—
537	#106	C₂³×C₃₁	1	4	yes	—
538	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
539	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
540	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
541	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
542	#103	C₂³×C₃₁	1	4	yes	—
543	#103	C₂³×C₃₁	1	2	yes	—
544	#103	C₂³×C₃₁	1	2	yes	—
545	#103	C₂³×C₃₁	1	4	yes	—
546	#103	C₂³×C₃₁	1	4	yes	—
547	#103	C₂³×C₃₁	1	4	yes	—
548	#102	C₂³×C₃₁	1	2	yes	—
549	#102	C₂³×C₃₁	1	2	yes	—
550	#102	C₂³×C₃₁	1	4	yes	—
551	#102	C₂³×C₃₁	1	4	yes	—
552	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
553	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
554	#97	C₂³×C₃₁	1	2	yes	—
555	#97	C₂³×C₃₁	1	4	yes	—
556	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
557	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
558	C₄×C₈	C₂³×C₃₁	1	4	yes	—
559	C₃₁	#61916	1	1	yes	characteristic, direct product factor, Sylow
560	#41	—	4	4	no	—
561	#41	—	8	2	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	2	4	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	4	2	no	—
567	C₂×Q₈	—	8	4	no	—
568	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
569	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
570	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
571	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
572	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
573	C₂×D₄	—	2	2	no	—
574	C₂×D₄	—	2	2	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
580	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	2	no	—
584	C₂²×C₄	—	2	2	no	—
585	C₂²×C₄	—	2	4	no	—
586	C₂²×C₄	—	2	4	no	—
587	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
588	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
589	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
590	Q₁₆	—	4	4	no	—
591	Q₁₆	—	4	4	no	—
592	SD₁₆	—	4	4	no	—
593	SD₁₆	—	4	4	no	—
594	SD₁₆	—	4	2	no	—
595	SD₁₆	—	4	2	no	—
596	D₈	—	4	4	no	—
597	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
598	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
599	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	2	4	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	2	2	no	—
612	C₄⋊C₄	—	2	4	no	—
613	C₄⋊C₄	—	2	4	no	—
614	C₄⋊C₄	—	2	4	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
617	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
618	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
619	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
620	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
621	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
622	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
623	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
624	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
625	C₂²⋊C₄	—	2	2	no	—
626	C₂²⋊C₄	—	2	2	no	—
627	C₂²⋊C₄	—	2	4	no	—
628	C₂²⋊C₄	—	2	4	no	—
629	C₂²⋊C₄	—	2	4	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	4	no	—
633	C₂²⋊C₄	—	2	4	no	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	4	no	—
636	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
637	C₄²	—	2	2	no	—
638	C₄²	—	2	2	no	—
639	C₄²	—	2	4	no	—
640	C₄²	—	2	2	no	—
641	C₄²	—	2	4	no	—
642	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
643	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
644	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
645	C₄²	C₂⁴×C₃₁	1	4	yes	—
646	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
647	C₂³	—	2	2	no	—
648	C₂³	—	2	2	no	—
649	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
650	Q₈	—	2	4	no	—
651	Q₈	—	2	2	no	—
652	Q₈	—	2	4	no	—
653	Q₈	—	2	4	no	—
654	Q₈	—	2	2	no	—
655	Q₈	—	4	2	no	—
656	Q₈	—	4	2	no	—
657	Q₈	—	4	2	no	—
658	Q₈	—	8	4	no	—
659	Q₈	—	8	1	no	—
660	D₄	—	2	2	no	—
661	D₄	—	2	4	no	—
662	D₄	—	2	2	no	—
663	D₄	—	4	2	no	—
664	D₄	—	4	4	no	—
665	D₄	—	4	2	no	—
666	D₄	—	8	1	no	—
667	D₄	—	8	1	no	—
668	D₄	—	8	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	4	no	—
679	C₂×C₄	—	4	2	no	—
680	C₂×C₄	—	4	4	no	—
681	C₂×C₄	—	8	4	no	—
682	C₂×C₄	—	8	4	no	—
683	C₂×C₄	—	8	1	no	—
684	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
685	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
686	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
687	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
688	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
689	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
690	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
691	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
692	C₈	—	2	4	no	—
693	C₈	—	2	2	no	—
694	C₈	—	2	2	no	—
695	C₂²	—	2	1	no	—
696	C₂²	—	2	1	no	—
697	C₂²	—	2	1	no	—
698	C₂²	—	4	2	no	—
699	C₂²	—	4	2	no	—
700	C₂²	—	4	2	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
705	C₄	—	2	1	no	—
706	C₄	—	2	1	no	—
707	C₄	—	2	4	no	—
708	C₄	—	2	1	no	—
709	C₄	—	2	1	no	—
710	C₄	—	2	4	no	—
711	C₄	—	4	2	no	—
712	C₄	—	4	2	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	2	no	—
716	C₄	—	4	2	no	—
717	C₄	C₃₁×#577	1	2	yes	—
718	C₄	C₃₁×#585	1	2	yes	—
719	C₂	—	4	1	no	—
720	C₂	—	8	2	no	—
721	C₂	—	8	2	no	—
722	C₂	C₃₁×#3487	1	1	yes	characteristic
723	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
724	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3214	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3550	C₂	semidirect
2. C₃₁×#3507	C₂	semidirect
3. C₃₁×#3401	C₂	semidirect
4. C₃₁×#3398	C₂	semidirect
5. C₃₁×#3394	C₂	semidirect
6. C₃₁×#3373	C₂	semidirect
7. C₃₁×#3369	C₂	semidirect
8. C₃₁×#3359	C₂	semidirect
9. C₃₁×#3225	C₂	semidirect
11. C₃₁×#3210	C₂	semidirect
12. C₃₁×#3125	C₂	semidirect
13. C₃₁×#3113	C₂	semidirect
14. C₃₁×#3103	C₂	semidirect
15. C₃₁×#2985	C₂	semidirect
284. #61916	C₃₁	direct (N × H)
333. #3550	C₂×C₃₁	semidirect
334. #3507	C₂×C₃₁	semidirect
335. #3401	C₂×C₃₁	semidirect
336. #3398	C₂×C₃₁	semidirect
337. #3394	C₂×C₃₁	semidirect
338. #3373	C₂×C₃₁	semidirect
339. #3369	C₂×C₃₁	semidirect
340. #3359	C₂×C₃₁	semidirect
341. #3225	C₂×C₃₁	semidirect
343. #3210	C₂×C₃₁	semidirect
344. #3125	C₂×C₃₁	semidirect
345. #3113	C₂×C₃₁	semidirect
346. #3103	C₂×C₃₁	semidirect
347. #2985	C₂×C₃₁	semidirect
559. C₃₁	#61916	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3550	2
C₃₁×#3507	2
C₃₁×#3401	2
C₃₁×#3398	2
C₃₁×#3394	2
C₃₁×#3373	2
C₃₁×#3369	2
C₃₁×#3359	2
C₃₁×#3225	2
C₃₁×#3214	2
C₃₁×#3210	2
C₃₁×#3125	2
C₃₁×#3113	2
C₃₁×#3103	2
C₃₁×#2985	2
#61916	31

Subgroups — C31×#61916 #21518675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61916 #21518675